The Lorenz mapping is a discretization of a pair of differential equations. It illustrates the pertinence of computational chaos. We describe complex dynamics, bifurcations, and chaos in the map. Fractal basins are displayed by numerical simulation.
We consider some dynamical properties of two-dimensional maps having an inverse with vanishing denominator. We put in evidence a link between a fixed point of a map with fractional inverse and a focal point of this inverse.
In this paper, we study the appearance, evolution and neighbourhood of two attractors of a dynamical system defined by a quadratic polynomial map 2 2The first is a Cantor-type attractor located on an invariant straight line. Thus, it suffices to study the restriction of the map T to this invariant line. The second is a closed curves cycle of period 2. We show, by a numerical approach, that when a parameter of the system varies, the evolution of the orbits in the region close to this second attractor is dependent on the evolution of the stable and unstable sets (homoclinic tangency) of a saddle cycle of period 2 located in this region.
Our study concerns global bifurcations occurring in noninvertible maps, it consists to show that there exists a link between contact bifurcations of a chaotic attractor and homoclinic bifurcations of a saddle point or a saddle cycle being on the boundary of the chaotic attractor. We provide specific information about the intricate dynamics near such points. We study particularly a two-dimensional endomorphism of (Z\ - Z$ - Z\) type. We will show that points of contact, between boundary of the attractor and its basin of attraction, converge toward the saddle point or the saddle cycle. These points of contact are also points of intersection between the stable and unstable invariant manifolds. This gives rise to the birth of homoclinic orbits (homoclinic bifurcations).
In this work, we consider some dynamical properties and specific contact bifurcations of two-dimensional maps having an inverse with vanishing denominator. We introduce new concepts and notions of focal points and prefocal curves which may cause and generate new dynamic phenomena. We put in evidence a link existing between basin bifurcation of a map with fractional inverse and the prefocal curve of this inverse.
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